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We show that $\\mathcal{F}(\\lambda)$ can be defined by a finite set of forbidden subgraphs if and only if $\\lambda < \\lambda^* := \\sqrt{2+\\sqrt{5}} \\approx 2.058$ and $\\lambda \\not\\in \\{\\alpha_2, \\alpha_3, \\dots\\}$, where $\\alpha_m = \\beta_m^{1/2} + \\beta_m^{-1/2}$ and $\\beta_m$ is the largest root of $x^{m+1}=1+x+\\dots+x^{m-1}$. 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